Divisor class groups of the semigroups of the highest weights (Q1335095)

Let \(S\) be a (commutative, with a unit and a torsion-free total quotient group \(\langle S\rangle\)) semigroup. Then denote by \(D(S)\) the set of divisorial fractional ideals, by \(\text{Prin}(S)\) the set of principal fractional ideals of \(S\) and by \(Cl(S) = D(S) /\text{Prin}(S)\) the divisor class semigroup of \(S\). Let \(G\) be a connected complex semisimple algebraic group, \(B\) be a Borel subgroup of \(G\), \(T \subset B\) be a maximal torus, \({\mathcal H} = {\mathcal H}(T)\) be the group of rational characters of \(T\) written additively, \(V = {\mathcal H}\otimes_ Z Q\), \(P \subset V\) be the weight lattice of \(G\), \(\{\omega_ 1, \dots, \omega_ r\}\) be the system of fundamental weights of \(P\) with respect to \(B\), \(P_ + = \mathbb{N} \omega_ 1 + \dots + \mathbb{N} \omega_ r\) be the semigroup of the dominant weights, \({\mathcal H}_ + = P_ + \cap {\mathcal H}\). The fundamental group of \(G\) is denoted by \(\pi_ 1(G)\). The main result of the paper is the Theorem: Let \(\pi^ 0_ 1(G)\) be the subgroup of \(\pi_ 1 (G)\) generated by all distinguished elements. Then \(Cl({\mathcal H}_ +) \cong \pi_ 1(G) / \pi^ 0_ 1(G)\).